Theorem lautcnv 37253

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Hypothesis

Ref	Expression
lautcnv.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautcnv	⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Proof of Theorem lautcnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2820	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautcnv.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 37248	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4		f1ocnv 6608	. . 3 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	3, 4	syl 17	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6		eqid 2820	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
7	1, 6, 2	lautcnvle 37252	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
8	7	ralrimivva 3186	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
9	1, 6, 2	islaut 37246	. . 3 ⊢ (𝐾 ∈ 𝑉 → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
10	9	adantr 483	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
11	5, 8, 10	mpbir2and 711	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3133   class class class wbr 5047  ^◡ccnv 5535  –1-1-onto→wf1o 6335  ‘cfv 6336  Basecbs 16461  lecple 16550  LAutclaut 37148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7442

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3012  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3935  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7140  df-oprab 7141  df-mpo 7142  df-map 8389  df-laut 37152

This theorem is referenced by:  ldilcnv  37278